Intermediate Algebra (11th edition)

OBJECTIVE 4 Solve an equation that is quadratic in form by substitution.

A nonquadratic equation that can be written in the form

for and an algebraic expression u,is called quadratic in form.

Many equations that are quadratic in form can be solved more easily by defining

and substituting a “temporary” variable ufor an expression involving the variable in

the original equation.

Defining Substitution Variables

Define a variable u, and write each equation in the form

(a)

Look at the two terms involving the variable x, ignoring their coefficients. Try to

find one variable expression that is the square of the other. Since we can

define and rewrite the original equation as a quadratic equation.

Here,

(b)

Because this equation involves both and we choose

Substituting ufor gives the quadratic equation

. Here,

(c)

We apply a power rule for exponents (Section 5.1), Because

we define The original equation becomes

Here, NOW TRY

Solving Equations That Are Quadratic in Form

Solve each equation.

(a)

We can write this equation in quadratic form by substituting (See

Example 5(a).)

Let

Factor.

or Zero-factor property

or Solve.

or Substitute for u.

or Square root property

The equation , a fourth-degree equation, has four solutions,

* The solution set is abbreviated Each solution can be veri-

fied by substituting it into the original equation for x.

- 3, -2, 2, 3. 5 2,  36.

x^4 - 13 x^2 + 36 = 0

x=  2 x= 3

x^2 = 4 x^2 = 9 x^2

u= 4 u= 9

u- 4 = 0 u- 9 = 0

1 u- 421 u- 92 = 0

u^2 - 13 u+ 36 = 0 u=x^2.

1 x^222 - 13 x^2 + 36 = 0 x^4 = 1 x^222

x^4 - 13 x^2 + 36 = 0

u for x^2.

x^4 - 13 x^2 + 36 = 0

EXAMPLE 6

2 u^2 - 11 u+ 12 = 0. u=x1/3.

1 x1/3 22 = x2/3, u=x1/3.

1 am 2 n= amn.

2 x2/3- 11 x1/3 + 12 = 0

2 u^2 + 7 u+ 5 = 0 u= 4 x-3.

u= 4 x- 3. 4 x- 3

14 x- 322 14 x- 32 ,

214 x- 322 + 714 x- 32 + 5 = 0

u^2 - 13 u+ 36 = 0 u=x^2.

u= x^2 ,

x^4 = 1 x^222 ,

x^4 - 13 x^2 + 36 = 0

au^2 +bu+c= 0.

EXAMPLE 5

aZ 0

au^2 buc0,

516 CHAPTER 9 Quadratic Equations, Inequalities, and Functions

*In general, an equation in which an nth-degree polynomial equals 0 has ncomplex solutions, although some

of them may be repeated.

Don’t stop here.

NOW TRY
EXERCISE 5
Define a variable u, and write
each equation in the form

(a)

(b)

• 111 x+ 22 + 4 = 0

61 x+ 222

x^4 - 10 x^2 + 9 = 0

au^2 +bu+c=0.

NOW TRY ANSWERS

5. (a)
   (b)
   6 u^2 - 11 u+ 4 = 0

u=x+2;

u=x^2 ; u^2 - 10 u+ 9 = 0